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 Coin tossing (Posted on 2004-06-11)
I threw a coin n times, and never got three tails in a row. I calculated the odds of this event, and found out they were just about even; 50%-50%. How many times did I throw the coin?

A second question: what were the chances of having not gotten three heads in a row either?

 See The Solution Submitted by Federico Kereki Rating: 3.6667 (6 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 re: Markov chain solution - 2nd part | Comment 16 of 25 |
(In reply to Markov chain solution - 2nd part by Old Original Oskar!)

To make this method work, the HHH and TTT states (that we had gotten along the way three heads--or tails--in a row) need to be broken down further, and understood that they exclude the case(s) where both HHH and TTT have been achieved.  Explicitly,

• H means neither HHH nor TTT has been achieved and the sequence ends in H. (understood that the preceding toss is not H)
• HH means neither HHH nor TTT has been achieved and the sequence ends in HH.
• T means neither HHH nor TTT has been achieved and the sequence ends in T.
• TT means neither HHH nor TTT has been achieved and the sequence ends in TT.
• HHHH means HHH has been achieved, TTT has not been achieved and the sequence ends in any number of H's.
• HHHT means HHH has been achieved, TTT has not been achieved and the sequence ends in a single T.
• HHHTT means HHH has been achieved, TTT has not been achieved and the sequence ends in TT.
• TTTT means TTT has been achieved, HHH has not been achieved and the sequence ends in any number of T's.
• TTTH means TTT has been achieved, HHH has not been achieved and the sequence ends in a single H.
• TTTHH means TTT has been achieved, HHH has not been achieved and the sequence ends in HH.

Given the above definitions, the transition probabilities are:

p(i + 1, hhhh) = .5 * (p(i, hhhh) + p(i, hhht) + p(i, hhhtt)) + .5 * p(i, hh)
p(i + 1, hhht) = .5 * (p(i, hhhh))
p(i + 1, hhhtt) = .5 * p(i, hhht)
p(i + 1, hh) = .5 * p(i, h)
p(i + 1, h) = .5 * (p(i, t) + p(i, tt))
p(i + 1, t) = .5 * (p(i, hh) + p(i, h))
p(i + 1, tt) = .5 * p(i, t)
p(i + 1, tttt) = .5 * (p(i, tttt) + p(i, ttth) + p(i, ttthh)) + .5 * p(i, tt)
p(i + 1, ttth) = .5 * (p(i, tttt))
p(i + 1, ttthh) = .5 * (p(i, ttth))

The results come out:

`i    h     hh     hhhh   hhht   hhhtt     t     tt    tttt  ttth   ttthh 1 0.5000 0.0000 0.0000 0.0000 0.0000 0.5000 0.0000 0.0000 0.0000 0.0000 2 0.2500 0.2500 0.0000 0.0000 0.0000 0.2500 0.2500 0.0000 0.0000 0.0000 3 0.2500 0.1250 0.1250 0.0000 0.0000 0.2500 0.1250 0.1250 0.0000 0.0000 4 0.1875 0.1250 0.1250 0.0625 0.0000 0.1875 0.1250 0.1250 0.0625 0.0000 5 0.1563 0.0938 0.1563 0.0625 0.0313 0.1563 0.0938 0.1563 0.0625 0.0313 6 0.1250 0.0781 0.1719 0.0781 0.0313 0.1250 0.0781 0.1719 0.0781 0.0313 7 0.1016 0.0625 0.1797 0.0859 0.0391 0.1016 0.0625 0.1797 0.0859 0.0391 8 0.0820 0.0508 0.1836 0.0898 0.0430 0.0820 0.0508 0.1836 0.0898 0.0430 9 0.0664 0.0410 0.1836 0.0918 0.0449 0.0664 0.0410 0.1836 0.0918 0.044910 0.0537 0.0332 0.1807 0.0918 0.0459 0.0537 0.0332 0.1807 0.0918 0.045911 0.0435 0.0269 0.1758 0.0903 0.0459 0.0435 0.0269 0.1758 0.0903 0.045912 0.0352 0.0217 0.1694 0.0879 0.0452 0.0352 0.0217 0.1694 0.0879 0.045213 0.0284 0.0176 0.1621 0.0847 0.0439 0.0284 0.0176 0.1621 0.0847 0.043914 0.0230 0.0142 0.1542 0.0811 0.0424 0.0230 0.0142 0.1542 0.0811 0.042415 0.0186 0.0115 0.1459 0.0771 0.0405 0.0186 0.0115 0.1459 0.0771 0.040516 0.0151 0.0093 0.1375 0.0730 0.0385 0.0151 0.0093 0.1375 0.0730 0.038517 0.0122 0.0075 0.1292 0.0688 0.0365 0.0122 0.0075 0.1292 0.0688 0.036518 0.0099 0.0061 0.1210 0.0646 0.0344 0.0099 0.0061 0.1210 0.0646 0.034419 0.0080 0.0049 0.1130 0.0605 0.0323 0.0080 0.0049 0.1130 0.0605 0.032320 0.0065 0.0040 0.1054 0.0565 0.0302 0.0065 0.0040 0.1054 0.0565 0.030221 0.0052 0.0032 0.0980 0.0527 0.0283 0.0052 0.0032 0.0980 0.0527 0.028322 0.0042 0.0026 0.0911 0.0490 0.0263 0.0042 0.0026 0.0911 0.0490 0.026323 0.0034 0.0021 0.0845 0.0455 0.0245 0.0034 0.0021 0.0845 0.0455 0.024524 0.0028 0.0017 0.0784 0.0423 0.0228 0.0028 0.0017 0.0784 0.0423 0.022825 0.0022 0.0014 0.0725 0.0392 0.0211 0.0022 0.0014 0.0725 0.0392 0.021126 0.0018 0.0011 0.0671 0.0363 0.0196 0.0018 0.0011 0.0671 0.0363 0.019627 0.0015 0.0009 0.0621 0.0336 0.0181 0.0015 0.0009 0.0621 0.0336 0.018128 0.0012 0.0007 0.0573 0.0310 0.0168 0.0012 0.0007 0.0573 0.0310 0.016829 0.0010 0.0006 0.0529 0.0287 0.0155 0.0010 0.0006 0.0529 0.0287 0.015530 0.0008 0.0005 0.0488 0.0265 0.0143 0.0008 0.0005 0.0488 0.0265 0.014331 0.0006 0.0004 0.0451 0.0244 0.0132 0.0006 0.0004 0.0451 0.0244 0.013232 0.0005 0.0003 0.0416 0.0225 0.0122 0.0005 0.0003 0.0416 0.0225 0.012233 0.0004 0.0003 0.0383 0.0208 0.0113 0.0004 0.0003 0.0383 0.0208 0.011334 0.0003 0.0002 0.0353 0.0192 0.0104 0.0003 0.0002 0.0353 0.0192 0.010435 0.0003 0.0002 0.0325 0.0177 0.0096 0.0003 0.0002 0.0325 0.0177 0.009636 0.0002 0.0001 0.0300 0.0163 0.0088 0.0002 0.0001 0.0300 0.0163 0.008837 0.0002 0.0001 0.0276 0.0150 0.0081 0.0002 0.0001 0.0276 0.0150 0.008138 0.0001 0.0001 0.0254 0.0138 0.0075 0.0001 0.0001 0.0254 0.0138 0.007539 0.0001 0.0001 0.0234 0.0127 0.0069 0.0001 0.0001 0.0234 0.0127 0.0069`

Where for line 10:

States HHHH, HHHT and HHHTT have probabilities 0.1807, 0.0918, 0.0459, adding up to .3184, which is the total probability that HHH will have been achieved, but not TTT. The same probabilities hold for TTT but not HHH.

The probability that nothing (neither TTT nor HHH) has been achieved is the total probability of states H, HH, T and TT: 0.0537, 0.0332, 0.0537, 0.0332, adding up to  .1738.

These probabilities, .3184, .3184 and .1738 add to .8106, leaving .1894 as the probability of both TTT and HHH having been accomplished. The .3184 + .1894 for a given TTT or HHH gives .5078 as the probability of each of those, disregarding whether the other has been achieved.  The .1738 of having achieved neither, divided by the 1-.5078 probability of not having achieved TTT gives the .3531 probability of having had neither, given not having the one.

Corrected typos.

Edited on June 13, 2004, 2:18 pm
 Posted by Charlie on 2004-06-13 13:49:52

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